True where it was derived, applied where it was not

“Methane is tetrahedral because carbon's orbitals hybridise into four equivalent sp³ bonds.”

The twenty-four operations are generated from the four coordinates, and the hydrogen 1s functions reduce to a₁ ⊕ t₂. Two ionisation energies are therefore required by symmetry, and the photoelectron spectrum shows bands at 12.7 and 23 eV. Four equivalent orbitals would give one band.

Tested in Hybridisation does not explain · the series on hybrids

“A hypervalent atom carries more than eight electrons.”

Counted honestly through the three-centre four-electron scheme, two electrons occupy a bonding orbital spread over three centres and two occupy an orbital that is non-bonding and sits on the ligands. The central atom never exceeds an octet.

Tested in Hypervalency without d orbitals · the series on hypervalency

“A molecule has no dipole moment because its bond vectors cancel.”

The exact statement is that a permanent dipole is possible only in C1, Cs, Cn and Cnv. The group is recovered from the coordinates and the polarity read off it, for every molecule drawn here — the cancelling-vectors argument gets the easy cases right by a route that does not generalise.

Tested in Symmetry forbids a dipole · the series on point group

“Deuterating a molecule lowers every X–H frequency by a factor of √2.”

Exact for a diatomic with an infinitely heavy partner and for nothing else. Computed for water the three ratios are 1.3667, 1.3868 and 1.3648 against √2 = 1.4142 — all short, by between 2 and 3.4 per cent. The one mode in this collection that does reach √2 is methane's doubly degenerate bend, at 1.4137, and it reaches it because the carbon does not move in that mode at all.

Tested in The isotope shift is arithmetic · the series on normal mode

“A band is 2zβ wide, where z is the coordination number.”

That is the width for a hypercubic connectivity and it fails for others: a Bethe lattice of the same coordination z has a band of width 4β√(z−1), which for z = 4 is 6.93β against the formula's 8β. The quantity that really is z, at every size and for every structure, is the second moment rather than the width.

Tested in The width of a band is a count of neighbours · the series on Bands in a solid

“Crystal field theory works, so the splitting really is electrostatic repulsion between ligand charges and d electrons.”

The point-charge model reproduces every ratio here exactly and gets the magnitude wrong by a large factor with any reasonable choice of charge and distance, while an angular overlap model with no charges in it at all reproduces the same ratios. What both models share is the geometry, so the geometry is what the ratios are about. The ordering of real ligands by splitting is settled in the spectrochemical series essay, and it is not an ordering by charge.

Tested in Two models, one ratio · the series on ligand field

“A correlation table lets any band in a spectrum be assigned to a bond.”

It works where the mode is localised and fails where it is not, and which is which is computable. Methane has no localised mode at all — its four stretching modes are a quarter in each of four bonds and its five bending modes a sixth in each of six angles — so no band in its spectrum can be assigned to a bond however good the table.

Tested in Group frequencies, and where they stop · the series on normal mode

“The eighteen-electron rule holds for transition-metal complexes in the way the octet rule holds for the second row.”

The octet rule follows from four valence orbitals all of which are strongly involved in bonding. The eighteen count relies on three metal orbitals being non-bonding, which is true only for ligands with no π interaction; a π donor pushes those three up and makes counts below eighteen ordinary, while a weak field can leave the antibonding pair low enough that twenty electrons are stable, as in nickelocene.

Tested in Eighteen is a count · the series on electron count

“The electron is most likely to be found at the nucleus, where the wavefunction is largest.”

Three numbers, all computed here and all correct: |ψ|² is largest at r = 0, the radial distribution peaks at exactly 1 bohr, and the ninety per cent contour is at 2.66. The volume element grows as r², so the three questions have three answers.

Tested in Where the electron is · the series on orbital

“The C–O stretching frequency measures the strength of the metal–carbon bond.”

It measures the C–O force constant. The correlation with metal–ligand bonding runs through back-donation, which weakens C–O while strengthening M–C, so the two move oppositely; and a stretching frequency is not a bond dissociation energy in any case, as the isoelectronic series here shows by varying only the charge on the complex while the metal changes as well.

Tested in Back-bonding is two interactions · the series on electron count

“Spreading electrons over more centres lowers their energy.”

Cyclobutadiene is delocalised in exactly the sense benzene is, and its delocalisation energy against a stated reference of isolated double bonds comes out exactly zero, where benzene's comes out exactly 2β.

Tested in Delocalisation is stabilising, and other things that are false in general · the series on aromaticity

“Surface effects vanish in the thermodynamic limit, so they can be ignored for a large sample.”

They vanish as a fraction of the whole, which is not the same as being negligible for the question asked. The two ends of a chain cost about 0.71β in total however long the chain is; that quantity does not shrink at all, and any measurement dominated by the ends — catalysis, adsorption, luminescence from a nanocrystal — is measuring the part that never converges.

Tested in Where a molecule stops being one · the series on Bands in a solid

“The π bond is the weaker half of a double bond, worth about 266 kJ/mol.”

That number is a subtraction, 614 − 348, between two bonds of different lengths: C=C is 1.34 Å and C–C is 1.54. The σ component of a double bond is not the same thing as a single bond, so the difference is not the π bond's energy — it is the difference between two whole molecules.

Tested in A double bond is not two single bonds · the series on overlap

“The trans influence explains why substitution happens faster at the position trans to a strong donor.”

Those are two different claims and the standard vocabulary separates them: the trans influence is a ground-state structural effect, measurable as a bond length in a crystal that is not reacting, while the trans effect is a statement about rates and therefore about a transition state. They correlate and are not the same ordering. The overlap argument computes bond orders in a ground state and says nothing about kinetics.

Tested in The trans influence is an overlap argument · the series on overlap

“Multicentre bonding is an exotic special case.”

It is the same three-by-three eigenvalue problem that describes the allyl radical, run with different diagonal entries. The two-centre bond is the special case — the one where the matrix happens to be two by two — and nothing in the arithmetic marks it out as the normal one.

Tested in Three-centre bonding, computed · the series on hypervalency

“VSEPR explains molecular shape.”

It predicts, exactly and with no free parameter, the arrangements of equivalent domains — 109.4712° for four, 90 and 120 for five. Every angle it gets from the lone-pair clause carries a fitted weight that no measurement supplies, so those predictions are fits with one point each.

Tested in What a lone pair is worth · the series on VSEPR

“Two orbitals happening to have the same energy is a degeneracy like any other.”

A degeneracy the group permits is stable — it survives any perturbation that keeps the symmetry. One the group does not permit is accidental, and any perturbation whatever splits it. The two behave completely differently under exactly the circumstances a chemist cares about.

Tested in Degeneracy is a group theorem · the series on representation

“Cyclobutadiene is a square molecule with a triplet ground state, as Hückel theory predicts.”

Hückel predicts that for a square, and the square is not stable. Its two electrons sit in a degenerate non-bonding pair, so its π energy falls linearly in the bond alternation while any elastic resistance rises quadratically — measured here by halving the step, which leaves the difference quotient unchanged for a linear dependence and doubles it for a quadratic. A linear gain beats a quadratic cost at small enough distortion, always.

Tested in The vibration that lowers the symmetry · the series on aromaticity

“VSEPR works because electrons repel electrostatically.”

Every angle VSEPR gets right for four, five and six domains is unchanged when the repulsion is made to go as 1/r², 1/r³, 1/r⁶ or 1/r¹². Those predictions do not depend on the force being electrostatic, or on its being a force at all — they follow from the arrangement being the maximally symmetric one.

Tested in Which angles are symmetry and which are the model · the series on VSEPR

“The spin-only formula is an approximation that works when the measurement is not very accurate.”

Its errors are one-sided. Across nine ions the measured moment is never meaningfully below the counted one and exceeds it by up to 0.93 Bohr magnetons for cobalt(II) — a pattern that identifies a term left out rather than scatter. The missing term is the orbital contribution, which can only add, and it is largest for the configurations whose ground state has orbital degeneracy.

Tested in A moment counts electrons, not orbitals · the series on magnetism

“Correlation tables are reference data.”

They are the result of restricting each parent character to the surviving classes and reducing it in the subgroup — one sum per representation, with the same whole-number and rebuild checks any other reduction obeys. Every row of the three tables here was computed rather than copied.

Tested in Descent in symmetry · the series on point group

“The force constant of a bond can be obtained from the frequency of its stretching band.”

True for a diatomic, where one frequency and one constant are the whole problem. For water there are three frequencies and four constants: fixing the stretch–stretch interaction anywhere between −0.30 and −0.10 mdyn Å⁻¹ and refitting the other three reproduces all three frequencies to better than a hundred-thousandth, with the O–H constant moving by 0.2 and the bending constant by 37 per cent.

Tested in The force field is not in the spectrum · the series on normal mode

“Hydrogen peroxide is achiral, because its enantiomers interconvert.”

The interconversion is a fact about a barrier, measured at about 4.6 kJ/mol over the anti arrangement and 29.5 over the eclipsed one, and neither is computed here. The point group of the equilibrium structure is C2 and the structure is chiral; whether the two hands can be separated at a given temperature is a different question with a different kind of answer, and conflating them makes chirality depend on the apparatus.

Tested in One coordinate, three point groups · the series on dipole

“A higher stretching frequency means a stronger bond.”

Sulfur dioxide's fitted S–O constant is 10.15 mdyn Å⁻¹ against water's O–H constant of 8.46, and its stretching bands are at 1,168 and 1,382 wavenumbers against water's 3,833 and 3,943. Ordering six molecules by force constant and by stretching frequency gives two different orders.

Tested in The frequency is not the bond strength · the series on normal mode

“The stretching frequency of a bond is √(k/μ) with μ the reduced mass of the two atoms.”

That formula is a diatomic and gives one number. Carbon dioxide's two C=O bonds share a force constant of 16.0 mdyn Å⁻¹ and a reduced mass of 6.86, from which the formula gives 1,991 wavenumbers; the molecule's two stretching modes are at 1,354 and 2,396, one 32 per cent below the estimate and one 20 per cent above.

Tested in The frequency is not the bond strength · the series on normal mode

“The Hückel pairing theorem holds for any alternant system.”

Pyridine's graph is alternant — the ring is bipartite, six-membered, and the colouring succeeds. Its levels are 2.107, 1.167, 1.000, −0.841, −1.000, −1.934, symmetric about nothing. Pairing needs every diagonal entry equal, which is a property of the matrix and not of the graph.

Tested in Hückel with a heteroatom · the series on delocalisation

“Sulfur hexafluoride needs d orbitals, because six bonds require six orbitals and sulfur's valence shell has only four.”

The premise is a symmetry statement and it is right: the six ligand σ functions span a₁g ⊕ eg ⊕ t₁u, and sulfur's 3s and 3p span a₁g and t₁u only, so the eg pair has no s or p partner. The conclusion does not follow, because a combination with no partner is non-bonding rather than impossible — it holds a pair of electrons on the ligands. Four bonding orbitals and two non-bonding ones account for all twelve electrons with no d orbital used.

Tested in Six bonds and four orbitals · the series on hypervalency

“A twisted ring is a different molecule and the comparison is not meaningful.”

The two matrices differ in exactly one entry and its sign, and the magnitude of that entry is unchanged — the overlap between two p orbitals turned half a turn apart has the same size and the opposite sign. What is not computed here is what the twist costs in bent bonds, which is a geometry the ring may not be able to afford; that cost is named and quoted rather than folded into the comparison.

Tested in The ring with a twist in it · the series on aromaticity

“Antiferromagnetic coupling means the electrons' spins repel each other.”

Nothing in the Hamiltonian refers to spin. The antiparallel arrangement is lower because it alone permits virtual hopping onto the neighbouring site — parallel spins are forbidden from doing so by the Pauli principle, which the computation shows as a triplet energy of exactly zero at every U, to twelve decimal places. The coupling is an accounting consequence of what hopping is allowed, not a force between spins.

Tested in What couples two spins · the series on magnetism

“A photoelectron spectrum shows the orbital energies of a molecule.”

It shows ionisation energies — differences between the energy of the molecule and the energies of its ion's states. Equating those with orbital energies is Koopmans' approximation, which assumes the remaining electrons do not relax and neglects the change in correlation energy. Both are of order an electronvolt for a valence orbital and they have opposite signs, so the agreement is a cancellation rather than a theorem.

Tested in What a photoelectron spectrum measures · the series on photoelectron

“A forbidden transition does not happen.”

Forbidden means the integral vanishes for the stated symmetry, and a vibrating molecule does not have that symmetry at every instant. Computing the product with a vibration included shows that the two odd vibrational species of an ML6 complex, T1u and T2u, make the transition allowed, while all three even species leave it forbidden — so the parity of the vibration is what decides, and the intensity that results is proportional to how much odd vibration is present.

Tested in Why a d–d band is weak · the series on representation

“The shape of an orbital is a physical property of the electron in it.”

It is a property of a function, and which functions the shell is written in is a choice. Any unitary mixture of the three 2p functions is another set of three, with lobes pointing elsewhere, and the filled shell's density is identical. The one thing that survives every choice is the sum, and the sum has no shape.

Tested in A filled shell has no shape · the series on contour

“Cyclopropane's carbons are sp³, with bent bonds accounting for the strain.”

Two quarter-s hybrids at 60° overlap by 0.625 — five eighths of their length. The bent-bond picture is the repair for exactly this, and it works by letting the orbitals point away from the internuclear lines so that the interorbital angle is near 104° while the ring angle stays at 60°. Calling the result sp³ describes the s content and not the directions, and the two are what the label is usually taken to fix together.

Tested in Hybrids that were never orthogonal · the series on hybrids

“The eighteen-electron rule counts electrons in the d shell plus the ligand donations.”

It counts filled orbitals. Nine orbitals — one s, three p, five d — filled twice over is eighteen, and every valence electron in the complex has to be in one of them. Reading it as a d-shell count makes the rule look like a coincidence about ten plus eight, and offers no way at all to see why a square plane stops at sixteen.

Tested in Sixteen is also a count · the series on electron count

“Water has two equivalent lone pairs, in two equivalent sp³ hybrid orbitals.”

As a basis it is legitimate and as a claim about states it is not. Water's four occupied valence orbitals belong to four different symmetry species — 2a₁, 1b₂, 3a₁ and 1b₁ — and its photoelectron spectrum shows four distinct bands at 12.6, 14.7, 18.5 and 32.2 electronvolts. Two of the four are the ones the equivalent-pair picture says should coincide; they are 2.1 electronvolts apart.

Tested in Water's lone pairs are not a pair · the series on photoelectron

“Fluorine is a poor pi donor to carbon because its 2p orbitals are small and overlap badly.”

Fluorine's 2p orbitals are the same principal shell as carbon's and overlap it perfectly well — a direct integration gives a 2p–2p pi overlap at the C–F distance comparable with the C–C value. What is poor is the match: fluorine's 2p sits several electronvolts below carbon's, so the gap in the denominator is large and the interaction is small however good the numerator is.

Tested in Overlap is not interaction · the series on overlap

“Puckering lets a ring reach the tetrahedral angle.”

Puckering always lowers the bond angle and never raises it, because a ring that leaves the plane spends more total curvature than a planar one and the angles have to pay for it. Puckering helps only for rings whose planar angle is above 109.5° — six atoms and more. Cyclobutane and cyclopentane pucker for a different reason and their angles fall when they do.

Tested in The angle a ring cannot have · the series on strain

“This kind of calculation needs a proper basis set to be worth anything.”

One variational parameter takes the bond length from 25 per cent wrong to right within a sixth of a per cent, and the binding from 63 per cent of the exact value to 84 per cent. What a larger basis buys after that is real and is a smaller fraction of the total than the first parameter bought. The failure of the fixed-exponent calculation is not a shortage of functions; it is one number held at the wrong value.

Tested in The atom does not bring its own orbital · the series on orbital

“Correlation energy is the difference between a Hartree–Fock energy and the exact one, so a system with no correlation energy has uncorrelated electrons.”

The definition names the part a determinant misses and not the part it captures. A determinant already contains the whole exchange hole, which for the two-site model at U = 0 is the entire correlation there is. Calling the rest 'the correlation' is a convention about a subtraction, and reading it as 'the amount by which electrons avoid each other' has the sign of the whole exchange contribution missing.

Tested in The hole that is not repulsion · the series on correlation

“Covalent bonding is the strongest kind of interaction between two units of matter.”

Per pair at contact it is not. A sodium and a chloride ion 2.8 Å apart are bound by 5.14 eV of Coulomb energy against β's 2.5 eV for a shared pair, computed from e²/4πε₀r. The comparison is the thing that is wrong — an ionic solid's binding is a lattice sum over every pair and a covalent solid's is not, so the two are not the same kind of quantity.

Tested in What holds a solid together · the series on cohesion

“Any degeneracy can occur in any molecule if the interactions are arranged to produce it.”

The permitted degeneracies are the dimensions in the group's own table and there are no others. The largest dimension in Oh is three, so an octahedral molecule cannot have four orbitals at one energy for any reason a symmetry argument would accept — and the sum-of-squares theorem is why the table has no larger entry: 1+1+4+9+9 twice is already 48.

Tested in Why a character table stops where it stops · the series on representation

“Orbital angular momentum is quenched because the crystal field is strong.”

It is quenched because the orbitals are real, and that has nothing to do with how strong anything is. Written in the five real d functions the operator Lz is i times a real antisymmetric matrix — dxz with dyz at one unit, dxy with dx²−y² at two, dz² with nothing — and an antisymmetric matrix has expectation exactly zero in any real vector whatever. Ten random real combinations give 10⁻¹⁵ or better.

Tested in An orbital carries no angular momentum · the series on magnetism

“Sulfur hexafluoride has six sulfur–fluorine bonds.”

It has six S–F contacts and less than six bonds' worth of bonding. In one three-centre four-electron system the computed bond order of each S–F link is 1/√2 = 0.7071, not the half the electron count suggests and not the one a line implies; the two together come to 1.414. Three such systems give twelve electrons round the sulfur and about four bonds' worth of bond order.

Tested in Hypervalency is about the ligands · the series on hypervalency

“A default isovalue is a reasonable convention, since any consistent choice is as good as another.”

Consistency in the isovalue is inconsistency in the claim. Solving for a stated enclosed fraction instead requires levels spanning a factor of 32 across the same seven orbitals — 3.9×10⁻² for the 1s down to 1.2×10⁻³ for the 4s — so the two conventions cannot both be sensible, and only one of them states what the picture means.

Tested in The isovalue nobody chose · the series on contour

“Crystal field theory explains the splitting in coordination complexes.”

It reproduces every geometric ratio exactly — four ninths for a tetrahedron, eight ninths for a cube, computed here to eight decimal places — and it cannot order the ligands, which is what the chemistry turns on. An angular overlap model with no charges in it orders all thirty-five comparable pairs correctly using one parameter whose sign says whether the ligand donates or accepts π density.

Tested in The spectrochemical series is not electrostatics · the series on ligand field

“The exponents in a standard basis set are physically meaningful parameters.”

They are the answer to a least-energy problem for one atom and they scale exactly with nuclear charge: the three optimised exponents for hydrogen are 0.151, 0.681 and 4.500, and for a one-electron ion of charge two they come out at exactly four times each, with an energy exactly four times as deep. Nothing in the optimiser knows that; it is the scaling of the Schrödinger equation, and it means an exponent carries a length rather than a property.

Tested in A Gaussian is the wrong shape · the series on basis

“The correction is circular, since the splitting is fitted to the same data it explains.”

One parameter is fitted and the shape it multiplies is computed, so the fit has 10 points and 3 parameters. And the fitted parameter is checkable: it comes out at 163 kJ/mol, or 13,600 cm⁻¹, against 7,800–13,900 cm⁻¹ measured optically for the same ions. Fitting the low-spin stabilisation pattern instead — which these aqua ions do not have — gives R² = 0.751, barely better than the bare line.

Tested in The double hump and what removes it · the series on ligand field

“The zero frequencies are an artefact and should be discarded without comment.”

They are the sharpest available check on a Hessian. A Hessian computed at a geometry that is not stationary, or built with an error in the transformation from internal to Cartesian coordinates, produces six numbers that are not zero and eigenvectors that are not rigid — and every downstream frequency then looks entirely reasonable. Discarding them without looking is discarding the diagnostic.

Tested in The six motions that are not modes · the series on normal mode

“VSEPR predicts molecular shape, so four ligands round a metal give a tetrahedron.”

A repulsion minimisation over four points on a sphere returns a tetrahedron for every central atom, because nothing in it depends on what the centre contains. Half of the four-coordinate complexes of nickel, palladium, platinum and gold are square planar, which has two 90° angles where a tetrahedron has none. The missing term is the ligand field stabilisation, computed here for every filling: largest at d⁸ and exactly zero at d⁰ and d¹⁰.

Tested in VSEPR does not reach a transition metal · the series on VSEPR

“A contracted basis set is a smaller basis set.”

It has the same primitives and therefore the same integrals; what it has fewer of is variational parameters. Six Gaussians contracted into one function and six left free give exactly the same energy for the atom the contraction was fitted to — to fifteen decimal places — and differ by 0.0283 hartree, seventy-four kilojoules a mole, at an effective charge of 1.238. Nothing about the basis changed between those two numbers except how many linear coefficients the calculation was allowed to choose.

Tested in A contraction is a decision made once · the series on basis

“A split-valence basis frees the diffuse functions because the valence region is where chemistry happens.”

The reason is arithmetic rather than chemical, and it holds for a change that pulls the orbital IN. What a frozen ratio gets wrong is how much tail it carries, and one free coefficient on the most diffuse primitive can subtract that tail. Freeing one coefficient at the diffuse end recovers 78 per cent of the loss at an effective charge of 1.238; freeing one at the tight end recovers 0.9 per cent.

Tested in A contraction is a decision made once · the series on basis

“A large electronegativity difference means a strongly polar bond, so Δχ can be used to estimate ionic character.”

Δχ is not comparable across scales without normalising, since the four run over ranges of 3.16, 7.99, 3.19 and 3.46 units from potassium to fluorine. After normalising, the O–H bond sits at 0.39 of the full range on Pauling, Allred–Rochow and Allen and at 0.045 on Mulliken. An ionic-character estimate built on the last would call water's bonds nearly non-polar.

Tested in A ranking is not a difference · the series on electronegativity

“The failure at six metals means one of the two rules is wrong.”

Rh₆(CO)₁₆ has 86 valence electrons. Taken as 18 per metal it leaves 11 pairs for 12 edges. Taken as 12 per metal it leaves 7 skeletal pairs for 6 vertices, which is n+1 — a closed deltahedron, and an octahedron is one. The boundary is where a cluster stops being a set of two-centre bonds and becomes a cage.

Tested in The bonds are what is left over · the series on electron count

“This is a subtlety about wavefunctions and does not affect ordinary calculations.”

It is the reason basis sets converged against total energies give poor dipole moments, why a method benchmarked on atomisation energies is not thereby benchmarked on charge distributions, and why the correlation energy recovered by a method is a poor guide to its density. Every one of those is the exponent-two-against-exponent-one behaviour showing up in practice.

Tested in A better energy is not a better answer · the series on approximation

“Electron deficiency means there are not enough electrons to go round.”

The tetramer has four occupied orbitals for eight framework atoms with eighteen contacts between them. Run a localising sweep over the four and every resulting orbital has amplitude on exactly four atoms — one carbon and the three metals of its own face, which was not put in anywhere. The same sweep on two separate ethenes returns orbitals on exactly two atoms with a participation number of 2.000.

Tested in Four centres, and the pair that will not localise · the series on multicentre

“Bredt's rule is about bridgeheads.”

A bridgehead double bond is trans in the largest ring containing both bridgeheads, so the question is only how large a ring must be to hold a trans double bond. The eight-ring threshold, read off a ring-closure calculation with no bridgehead in it, sorts nine bicyclic alkenes into isolable and not exactly as the record does — including adamantene, which has the same bicyclo[3.3.1] framework as an isolable compound and whose eight-ring is bridged shut, leaving six.

Tested in The double bond a ring cannot hold · the series on strain

“The centrifugal distortion constant is a fitted parameter.”

It is fitted in practice and it is predictable from a different experiment: D = 4B³/ω². For CO that gives 6.037×10⁻⁶ cm⁻¹ against a fitted 6.122×10⁻⁶; for HCl 5.088×10⁻⁴ against 5.319×10⁻⁴; for HF 2.149×10⁻³ against 2.151×10⁻³; for N₂ 5.737×10⁻⁶ against 5.760×10⁻⁶. Four molecules, three orders of magnitude in the quantity, and agreement within a few per cent.

Tested in The rotor that stretches · the series on rotation

“Overlap increases as two atoms approach.”

The 1s–2s overlap is −0.0158 at 0.5 bohr and −0.2820 at 4.17, a factor of eighteen in the wrong direction, and it falls again beyond. The 2p σ overlap is +0.973 at one bohr, passes through zero at 5.06, and is −0.319 at eight — larger in magnitude than it is at four.

Tested in Closer is not more overlap · the series on overlap

“The band width is what measures how strongly a site is bound.”

The width of these bands is exactly twice the coordination — 4, 8 and 12 — so it is proportional to Z, while the binding rises far more slowly. The width is a range and the binding is an integral over occupied levels, and the two agree only if the shape of the distribution between the edges is fixed. It is not: the chain's is edge-heavy and the cubic one's is centre-heavy.

Tested in The bond that weakens as neighbours multiply · the series on cohesion

“Fitting two parameters to two bands is not a test of anything.”

The two parameters are solved from the first two bands in closed form, and the position of the third is then a prediction. For the fluoride and the aquo complex, where a third band is measured, the predictions are 35,442 and 38,529 cm⁻¹ against measured 34,400 and 37,800 — within three per cent, on a band not used in the fit.

Tested in The electrons repel less inside the complex · the series on ligand field

“Hückel's rule comes out of diagonalising the adjacency matrix of a ring.”

The energies are α + β(ω^k + ω^−k) with ω = exp(2πi/n), which is α plus twice the real part of the character of the k-th representation of the cyclic group. Computing them that way and comparing against the Jacobi eigensolver agrees to better than 10⁻⁹ at every ring size from three to twelve. Nothing is diagonalised on the first route.

Tested in The ring's levels are its group's characters · the series on aromaticity

“The Möbius ring is an exception that needs a separate treatment.”

A twisted ring closes onto itself with a sign, so what acts on it is not a representation of the cyclic group but a projective one: the generator carries ω^(k+½) rather than ω^k. The levels become 2cos((2k+1)π/n), no representation lands on the real axis when n is even, the lowest level is a pair rather than a single, and every closed shell holds 4n electrons instead of 4n+2.

Tested in The ring's levels are its group's characters · the series on aromaticity

“The identity between the correlation energy's exponent and the bonding occupation generalises to larger systems.”

For two centres it holds to five figures at every repulsion. For four centres in a chain the exponent is 1.98540 at U = t while the largest occupation is 1.98378 — the exponent is above every occupation, so it is no average of them — and at U = 4t it is 1.78369 against a largest occupation of 1.78878, below it. It is neither an average nor a bound.

Tested in A third kind of correlation · the series on correlation

“A completely filled band means an insulator.”

Two systems with two electrons and two orbitals per site, so that the lower set is exactly full in both. With the two sets clear of one another the gap is 1.000 at 12, 24, 48, 96 and 192 sites — a fixed number. With them overlapping it is 0.098 at twelve sites and 0.0056 at a hundred and ninety-two, and falling. The electron count is identical in the two cases.

Tested in A full band is not an insulator · the series on metal

“A gem-dimethyl group also crowds the closed ring, which would oppose the acceleration.”

Not for the rings the measurements are on. A transannular contact is a pair four or more bonds apart, and a ring of five, six or seven has none — a counting fact with no parameter in it. From eight onwards the term exists and always costs: 3.31 kJ/mol at its best position in an eight-ring and 47.96 at its worst in a nine.

Tested in A ceiling that rises where the measurements fall · the series on strain

“A counterpoise correction improves the computed bond length.”

A correction recomputed at every separation does, by 4.456 millibohr here. A correction frozen at any single separation does not move it at all: the frozen curve is the uncorrected curve less a constant, so its minimum is the uncorrected minimum, to nine decimal places, at all seven reference geometries tested.

Tested in A correction computed at one length · the series on basis

“Hückel theory can be fitted to ionisation energies.”

It can be fitted to two of them, exactly, and to six of them no better than 0.456 eV — a floor set without fitting anything, because ethene and benzene have the same eigenvalue of exactly 1 and ionisation energies 1.27 eV apart, and butadiene and naphthalene share 0.618 and differ by 0.94.

Tested in A parameter that never finds a value · the series on models

“A small symmetry breaking is a small effect, so first-order perturbation theory describes it.”

The first-order splitting λ⟨1/r²⟩ is 0.15 per cent adrift from the exact one at λ = 0.0003 and 51 per cent adrift at λ = 0.1 — which is a quantum defect of 0.21, far smaller than any real alkali's.

Tested in How nearly a broken symmetry survives · the series on representation

“A count of edges is a cheap stand-in for a diagonalisation.”

It is exact on a bipartite net below a contrast of 1.5272, found by bisection, and wrong above it. On a triangular net it is already wrong at a contrast of 0.25, at two compositions out of three searched, because a net with odd rings has no arrangement in which every bond is unlike.

Tested in The arrangement a count cannot pick · the series on cohesion

“The field a neighbouring ion produces is a small perturbation on the one next to it.”

It is for four of the eight pairs here and not for the other four. The sulphide ion's ninety per cent surface is displaced by 36 per cent of its own radius in magnesium sulphide's field, which is not a small displacement and is reported as a failure of the expansion rather than as an answer.

Tested in The surface a neighbour moves · the series on contour

“A conclusion drawn from partial charges is only as good as the electronegativity table it used.”

Not when the conclusion is a ranking and the molecule has two elements. Every charge is then one number times a pattern fixed by the formula, a change of table changes only that number, and the rank correlation of infrared intensity against amplitude is identical on all four tables to machine precision for water, ammonia, methane, boron trifluoride and sulfur dioxide.

Tested in The table that could not have mattered · the series on dipole

“A partial-charge model can be checked by seeing whether it gives a molecule the right dipole.”

For methane and boron trifluoride it gives exactly zero on all four tables, because the point group forbids a dipole and no assignment of charges to symmetry-equivalent atoms can produce one. Agreement there is a check on the geometry, not on the charges.

Tested in The table that could not have mattered · the series on dipole

“Holding the alternation uniform is a small approximation.”

It is small where the chain is strongly dimerised and not where it is weakly dimerised. At an elastic constant of 2.6 the end bond alternates at 0.0883 against a bulk value of 0.0260, so the uniform assumption is wrong by more than a factor of three exactly where the end matters most.

Tested in The chain distorts hardest where it stops · the series on Peierls distortion

“A second measurement of a quantity already measured refines it.”

Not when the second is the same function of the same combination. The shapes of the two bands are both functions of t⊥/Δ, so one per cent measurements of both leave the coupling at 52 per cent and the separation at 51, against 1.13 and 1.40 for a pair that differs. The confidence region is 36 times larger.

Tested in Two ways of being second order · the series on Bands in a solid

“The longest ring separations in a molecule are the best measure of how a response decays.”

At the largest separation there is exactly one pair of rings, the two ends, and its entry is an end effect. Including it lengthens the fitted reach from 0.6385 to 0.7480 on the twelve-ring molecule — a seventeen per cent error, in the direction that makes every reach look longer.

Tested in A reach that has no length · the series on aromaticity

“Quoting a bend constant with an uncertainty describes what the spectrum determined.”

No combination of methane's six bend constants alone is fixed. The projection restricted to that block has eigenvalues 0.714 and five of 0.500, so every combination of them is partly traded against constants outside the block, and an uncertainty on one number cannot say that.

Tested in The forty-five that are fixed · the series on normal mode

“A length fitted over too short a window is corrected by using a longer chain.”

Fitted over twelve bonds the exponent is −0.550 on a chain of 96, −0.514 on one of 160 and −0.476 on one of 320. Making the chain three times longer moves it away from −1 rather than towards it, because the window and not the chain is what is short.

Tested in A decay that keeps slowing down · the series on Peierls distortion

“The near-tie bound applies to any predictor the exact one applies to.”

It needs a scale. VSEPR predicts an arrangement, and the difference between AX₄ and AX₃E is a label change rather than a distance, so a ratio of a measured difference to it means nothing. The exact instrument works there because equality is a question a label can answer.

Tested in The pair that is not a tie · the series on models

“The number of localised descriptions is a number.”

It is a number and it was reported as one before it was one. Ten was the answer after three dry batches, fourteen after 480 starts, and fourteen after 4,000 — with the last new description arriving at start 124 and nothing after it.

Tested in Where the count stops being an effort · the series on multicentre

“A coordinate set with sign changes in it cannot be counted by orbits.”

What matters is a sign change under an operation that fixes the coordinate, not a sign change anywhere. Ferrocene's coordinates suffer a hundred sign changes and all five of its orbits are symmetric, because the operations carrying the signs exchange the two rings.

Tested in A formula that predicts minus eleven vibrations · the series on point group

“The interhalogens are a test of whether the disagreement survives without hydrogen.”

They are all uncontested, but five of the six sit between 3.2 and 9.2 times the boundary — they were never candidates. Only Br–Cl comes close, at 1.07 times, and it is the closest agreed bond in the whole collection.

Tested in A dispute is a small difference · the series on dipole

“A discrepancy this small does not matter, being a thousandth of a hartree.”

It is 1.89 × 10⁻⁴ hartree, which is half a kilojoule a mole — small against a bond and comparable with the interaction energies counterpoise corrections are applied to. It is 15.7 per cent of the correction it is a correction to.

Tested in The assembly that counts one share twice · the series on basis

“A nearest-neighbour repulsion of half the on-site one is a mild extension of a Hubbard model.”

On this ring the alternating structure factor rises fastest at V = 0.5625 U, and at V = U/2 the double occupancy has already risen from 0.039 to 0.190 and the structure factor from 0.149 to 1.466. That is the middle of a crossover, not a perturbation of either end.

Tested in The give-back that turned into a saving · the series on correlation

“A decay length fitted over the interior pairs of an acene is a property of the interior.”

Excluding the single longest pair leaves every other pair that includes an end ring in the fit, and those are systematically high: on the linear six-ring molecule the spread within one separation class is already 1.520 and it grows with the molecule, to 1.596 at eight rings.

Tested in Neither of the two separations · the series on aromaticity

“The ceiling is a rate-like quantity that can be put on an Arrhenius plot beside a measured ratio.”

It curves and a rate ratio does not. At the difference that matches it best the margin traces 1.015, 1.004, 1.000, 1.003, 1.011, 1.021 across 253 to 373 K — a shallow U with its minimum at the reference temperature, which no straight line on an Arrhenius plot can produce.

Tested in One number decides which way it breaks · the series on strain

“Two bands built on different graphs are the general case, and one graph is the special case.”

One graph twice, on a triangular net, gives exponents of −1.105 and −1.105 — the two-graph behaviour exactly. What makes the square net special is that its spectrum is symmetric about its own centre, which has nothing to do with how many graphs there are.

Tested in The constant that belonged to one net · the series on Bands in a solid

“A σ orbital that sits 99.9 per cent on oxygen is a strongly polarised bond.”

It is not a bond at all. With the artefactual overlap the two-level problem barely couples, so the lower orbital is essentially oxygen's own hybrid; with the corrected overlap the share is 58.8 per cent, which is a polarised bond.

Tested in The node that decided a picture · the series on hybrids

“A model with one free parameter can be tuned to a trend it has the right shape for.”

Lowering the metal orbital multiplies the donor over-prediction from 1.17 to 12.9 and drives the acceptor under-prediction from 0.84 to 0.59. Both sides get worse together, so the free parameter has no useful value at all.

Tested in A denominator that fails both ways · the series on ligand field

“A ring of forty sites is large enough that its Peierls alternation is the infinite chain's.”

At K = 1.2 it is, to 0.0055 per cent. At K = 2.4 the same ring is 36.37 per cent above the infinite chain's value, and at K = 3.0 it is 127.58 per cent above it. Whether forty sites is large depends on the spring, because a stiffer spring gives a smaller gap and a longer coherence length.

Tested in The amplitude the collapse left behind · the series on metal

“A local search over arrangements is a weak instrument, so its answers should be treated as approximate everywhere.”

On sixteen sites the whole landscape has two local optima at both contrasts tested, and five random starts give a 99 per cent chance of the global one. The instrument is weak on thirty-six sites and is very nearly exact on sixteen; the weakness is a property of the problem's size, not of the method.

Tested in Twelve basins where there were two · the series on cohesion

“Missing the best arrangement costs little, because the energies of good arrangements are close together.”

On thirty-six sites at a contrast of four the second basin is 0.0068 per cent below the first. That is the reason missing it is cheap in energy and total in arrangement: the two are different structures reported as almost the same number, and the question being asked is which structure wins.

Tested in Twelve basins where there were two · the series on cohesion

“A set of levels that has become a band has a width set by the coupling between neighbours.”

The coupling between typical neighbours is 1.1 × 10⁻¹⁹ for runs of four and 3.6 × 10⁻¹⁷⁵ for runs of eight, at half concentration. A width computed from those numbers is not small; it is below anything that could be called a quantity.

Tested in A band that is a hundred and seventy decades of nothing · the series on defect

“A search that agrees with itself ninety-eight times in a hundred has found the answer.”

The cage with 98.3 per cent dominance is one whose five descriptions differ by two parts in a million. The agreement is real and it is agreement about a choice that does not matter, so it is evidence of nothing about the bonding.

Tested in Fifty descriptions of one molecule · the series on multicentre

“At the largest basis the assembly overshoots by more than half the correction, which is a serious error.”

At six Gaussians a centre the whole correction is 1.49 × 10⁻⁵ hartree, which is 107 times below a kilocalorie a mole. Fifty-three per cent of it is 7.9 × 10⁻⁶ hartree. A percentage quoted without the energy it is a percentage of is not a measure of harm.

Tested in One basis size where it is worth doing · the series on basis

“Bisection finds the crossing that is there.”

Given [2, 5] it finds no bracket at all, because two sign changes return the endpoints to the same side. Given [2, 4] it returns 3.8955 and given [3.9, 4.5] it returns 4.1595, and it reports both in the same form.

Tested in A sign change is not always a zero · the series on correlation

“The count and the repulsion are two measures of the same distortion, one stepped and one smooth.”

The repulsion rises monotonically by 4.20 per cent from the tetrahedron to the plane and is highest exactly where the count changes. So the one geometry the count singles out is the one the energy makes least likely, and the count says nothing whatever about the 99 per cent of the path in between.

Tested in A count that changes at one point · the series on hypervalency

“The residual test says in advance whether the extrapolation is allowed.”

It refuses half filling and it refuses quarter filling, in the same words. At half filling the residuals are 10⁻¹⁰ because the contrast is exactly constant; at quarter filling they are 10², because it is not converging at all. The test is a relative spread and its denominator is the departure it is measuring.

Tested in A ratio of exactly one is a tie · the series on photoelectron

“The tilted shell has six events at a general angle and four or five where two coincide.”

It has one avoided crossing, at every tilt from zero to eighty degrees, and two with the field exactly along x. The six are two-state estimates of where a coupling would match a gap; the exact five-level spectrum has no level crossing at all and exactly one interior minimum in any adjacent gap.

Tested in None of the six was a crossing · the series on representation

“The size of the margin says how strongly the mixed description is preferred.”

The margin is monotone in the s character of the outward hybrid and depends on nothing else — not on the effective charge, not on the bond length. Across three charges and three bond lengths the separation of the centroids moves by a factor of 1.71 and the criterion moves in its sixth decimal. The margin re-reports the hybrid it was given.

Tested in The five figures were an identity · the series on hybrids

“Which groups a collection can count in is a fact about group theory.”

It is a fact about which tables a calculation happens to hold. C3v is also of order six and was held; D2, D4 and D5d were held; D3 was not. Nothing distinguishes D3 mathematically. It is simply the group no earlier count had needed.

Tested in The group nobody wrote a table for · the series on hypervalency

“An exactness with no interaction behind it is fragile in a way one with an interaction is not.”

Both are symmetry statements and both go at once. What distinguishes them is what happens afterwards: here the level moves to the mean of the two site energies, which is a place first-order theory names in advance, so the exactness is replaced by a formula rather than by nothing.

Tested in A symmetry holds or it does not · the series on overlap

“A count that comes out of a reduction depends on which irreducible representations the reduction is against.”

Not this one. The σ set falls into four species in D3h, three in D3 and three in Oh, with different names and different dimensions in each. The number matched is 4 in all three and the count is 6 − 4 = 2 in all three. The labels are how the count is computed and not what it measures.

Tested in The count the table was hiding · the series on hypervalency

“The change in the correlation energy is a second diagnostic that would explain the composite's failure.”

It is the failure. The composite is the target's mean field plus the reference's correlation energy, so its error equals the reference's correlation energy minus the target's, identically. Across twenty points on two axes the largest departure between the two is 2.2 × 10⁻¹⁶.

Tested in The second number is the error, rearranged · the series on approximation

“A candidate diagnostic with no score is a candidate that was not ruled out.”

Two of the nine put no point from one axis within fifteen per cent of any point from the other, so the statistic has nothing to compare and returns nothing. The highest occupied level takes values from 0.019 to 0.151 on the repulsion axis and from 0.228 to 2.704 on the site-energy axis: disjoint ranges, and no evidence either way.

Tested in The second number is the error, rearranged · the series on approximation

“The truncation is therefore wrong.”

It is exact where the field is along z, and the two calculations agree there to every digit — 2.0876 × 10⁻⁵ from both. The error is zero at zero tilt and grows to 21 per cent at forty degrees, which is what a truncation that respects one direction and not the others does.

Tested in The variation was the basis · the series on representation

“A rank correlation of one half between two quantities means one is largely the other.”

The rank correlation here is 0.503 in magnitude and the variance accounted for is 0.029. The two statistics answer different questions, and only the second bears on whether a quantity has a simpler name.

Tested in A correlation is not an account · the series on electronegativity

“A rotor either way is a detail.”

The ceiling is a power law in the rotor count — each rotor multiplies it by 3.32 — so one rotor is a factor of three and a half and two rotors are a factor of eleven. In an argument whose tightest verdict was decided by ten per cent, that is the largest single uncertainty in the argument and it is not a measurement error.

Tested in The count that was never written down · the series on strain

“The even ring is the control that separates the ring from the frustration.”

It separates the ring from the frustration and leaves the parity of the count varying with both. The control chain had eight spins, which is even. Every case in the original sweep had frustration and odd parity aligned.

Tested in It was the count, not the frustration · the series on magnetism

“The closest pair in an input is the pair with the smallest difference in it.”

Measured that way — within a tenth of the input's own range — the test reports a capacity ratio of 43.9 for the capacity against itself, because the capacities span two orders of magnitude and a tenth of their linear range covers everything below potassium. Adjacency in rank is scale-free and returns 4.95.

Tested in The worst of the six was the one we asked about · the series on electronegativity

“The best boundary is a line, and quoting its slope states the rule.”

Forty-one of the 401 slopes tie at three mistakes, spanning 0.25 to 0.72 — nearly a factor of three. They classify all 153 pairs identically because the pairs are sparse where the lines differ. Quoting one slope reports the sweep's arithmetic rather than a property of the tables.

Tested in Two numbers caught what one could not · the series on dipole

“The exponent is flat past sixty bonds, so the far end could be anywhere beyond it.”

It is flat past each profile's own reach because there are no more local rates to add, so a window of eighty and one of a hundred and twenty fit exactly the same points and return the same number to every digit. That flatness is the window not being honoured, not the choice not mattering.

Tested in The other window was a plateau too · the series on Peierls distortion

“More bonds give a smaller exponent at a fixed spin count.”

At a 20–300 K window the four-spin exponents fall with every bond added, 0.287, 0.221, 0.214, 0.208. At 40–300 K they read 2.31, 0.70, 3.07, 1.83, with no order at all. The trend belonged to the window.

Tested in The sign rule holds between two poles · the series on magnetism

“Finding out how much of the ordering is the fit requires the fifth point.”

The cubic's coefficient is one sixth of a second difference of three measurements, so the shift that flips its sign is computable from the four points alone. Nitrogen flips on a change of 1.56 per cent in its second ionisation energy and sodium on 79 per cent, which predicts exactly which atoms a fifth point reaches before one is added.

Tested in Six of fifteen change verdict · the series on electronegativity

“Phosphine's computed splitting is zero.”

It is below what double-precision arithmetic can resolve between two eigenvalues of that size, which is 1.3 × 10⁻¹² wavenumbers, so the solve reports a bound. The action, which does not underflow, puts the true value near 2.5 × 10⁻¹⁶ — an inversion time of about a day against ammonia's twenty-one picoseconds.

Tested in It was never the mass · the series on inversion

“A radius cannot be brought into this comparison because radii are tabulated for other purposes and not on a scale all these atoms share.”

Not if it is computed rather than quoted. A hydrogenic orbital of the atom's valence shell at the effective charge Slater's rules give it has an exact mean radius and an exact root-mean-square radius, both closed forms in the quantum numbers and the charge, and both on one scale for every atom by construction.

Tested in A size the fit was not made from · the series on electronegativity

“The screening model −λ/r² is a small-defect approximation to a real alkali atom.”

Its s defect is ½ − √(¼ − 2λ), which cannot exceed one half at any strength. Lithium's 0.40 is reachable; sodium's 1.35, potassium's 2.18, rubidium's 3.13 and caesium's 4.05 are not reachable at all. Fitted to lithium's s defect, it puts the d defect at 0.048 against a measured 0.002.

Tested in Four alkalis the model cannot hold · the series on representation

“The counts agree, which shows the two criteria substantially concur.”

The totals are 37 and 39 of 48, which look like agreement and are partly two disagreements cancelling. Pair by pair they agree on 42. Comparing classifications by their totals is the same mistake as reading a count of descriptions as a measure of ambiguity.

Tested in A second criterion left a gap too · the series on multicentre

“Two sweeps that both leave a verdict standing are better evidence than one.”

Only if they explore different things. Here the union of the two ranges is 0.8068 in g/RT and the sum of their widths is 1.3874, so treating them as independent would credit the comparison with 72 per cent more coverage than it has. The temperature sweep added exactly nothing the gauche sweep does not contain.

Tested in Two sweeps and one lever · the series on strain

“A transposition that needs a correlation inside the unit interval is a transposition worth worrying about.”

Every one of the eight purchasable pairs is reachable at some correlation below one, so reachability separates nothing. The prices do: they run from 0.154 to 0.990, and a correlation of a sixth between two quantities from one literature is unremarkable while one of 0.99 is a different kind of claim. Reachability and cost are different questions and only the second orders the pairs.

Tested in Three chains and ninety orderings ruled out · the series on models

“A bonding orbital's momentum profile along the bond is zero at π/R and an antibonding orbital's is not.”

True for an s basis and false for p functions along the bond. For nitrogen's 2p functions at 2.074 bohr the σ bonding combination carries a sine and is at 0.834 of its own peak at π/R, while the σ antibonding combination carries the cosine and is exactly zero there. The factor is a cosine exactly when the orbital's inversion parity equals the atomic function's, on all six valence combinations.

Tested in The zero is a parity, not a bond · the series on orbital